manuscripta mathematica

, Volume 87, Issue 1, pp 435–448

Determination of the poles of the topological zeta function for curves

  • Willem Veys
Article

DOI: 10.1007/BF02570485

Cite this article as:
Veys, W. Manuscripta Math (1995) 87: 435. doi:10.1007/BF02570485

Abstract

Tof ∈ℂ[x1…,xn] one associates thetopological zeta function which is an invariant of (the germ of)f at 0, defined in terms of an embedded resolution of (the germ of)f−1{0} inf−1{0}. By definition the topological zeta function is a rational function in one variable, and it is related to Igusa’s local zeta function. A major problem is the study of its poles.

In this paper we exactly determine all poles of the topological zeta function forn=2 and anyf ∈ℂ[x1,x2]. In particular there exists at most one pole of order two, and in this case it is the pole closest to the origin. Our proofs rely on a new geometrical result which makes the embedded resolution graph of the germ off into an ‘ordered tree’ with respect to the so-callednumerical data of the resolution.

1991 Mathematics Subject Classification

14B05 14H20 32S45(11S40) 

Key words and phrases

Topological zeta function curve singularities resolution graph 

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • Willem Veys
    • 1
  1. 1.Department WiskundeLeuvenBelgium

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